Measure, Weight, and Number

When I tell people that my first love was not poetry but mathematics, they usually knit their brows and wonder what brought about that leap across a chasm.

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Augustin-Louis Cauchy1789-1857

It was no leap. Poetry is a world of meaning rendered in musical order, and mathematics is a world of number and shape, of momentum and direction, of unity and difference and relationship; it too has its streams and trees and branches and leaves, its planets gliding along realms of space curved like a water slide, its singularities at the heart of a bottom-less funnel, its surprises, its vistas without end. So God is often portrayed as a geometer, measuring out the world with a compass and saying to the waters, "Thus far shall thy proud waves be stayed."

Plato, that philosopher with the soul of a mystic, had an inscription carved over the archway of his Academy: "Let no one ignorant of geometry enter here." His sense was that meditation upon mathematical objects was a step along the way to the contemplation of the Good. That is because mathematical objects do not change or decay, and what we know about them, we know with certainty. The real existence of a single mathematical object — a number, a curve, an infinite series — is a knife in the heart of materialism. Real knowledge of such is a knife in the heart of sophism and skepticism.

So we shouldn't be surprised to find so many deeply devout men among the greatest of mathematicians. Blaise Pascal, inventor of probability theory, is one of the most penetrating theologians who ever lived. Leonhard Euler was the Johann Sebastian Bach of German mathematics in the 18th century: a devout Lutheran, happily marled, with a big brood of children, and possessed of immense creativity and intuition. Bernhard Riemann originally went to the university to become a pastor; his tombstone cites Saint Paul: All things work together for those who love God. Georg Cantor, the titan of set theory and discoverer of trans-finite numbers, believed that his work on the nature of infinity demonstrated the necessary existence of God. Kurt Gödel, inspired by Cantor's work, turned his own world-shaking Incompleteness Theorem to account, developing from it a revision of Saint Anselm's proof. Albert Einstein was once asked why he wanted to work at Princeton. "So that I can have lunch with Kurt Gödel," he said.

And there is one more man who ranks with the three or four greatest ever, whose fingerprints are to be found in every branch of mathematics: the devout Catholic, Augustin-Louis Cauchy (1789-1857).

Boy Genius

Augustin-Louis had advantages that are lacking to young men today. He was never subject to institutional chloroform: he did not go to school during his early years, but was educated by his father, himself a man of impressive learning. Then at the urging of the elder mathematician and astronomer Joseph-Louis Lagrange (1736-1813), he went to the École Centrale du Panthéon, to study — not math, but classical languages. At age sixteen he began his mathematical studies in earnest, at the École Polytechnique; ability and not age was his passport. His work was so brilliant that Napoleon, who by then had taken over the revolutionary regime — much to the benefit of the conservative Cauchy family — tabbed the young man to be one of his chief engineers at Cherbourg. Napoleon wanted to realize a long-held aim of the French, which was to dredge and enclose the harbor at Cherbourg, building "moles" and jetties and fortified hills to render Cherbourg more useful as a port and impregnable as a military redoubt. The project took many stages, but until recently Cherbourg boasted the largest artificial harbor in the world. Cauchy was nineteen when he began. What are our brightest sons doing at that age now — if their brains have not been soaked in drugs, and their souls in vice?

In his spare time Cauchy was writing papers filled with original work on solid geometry, calculus, and sound waves; I should note that a "paper" proving a subtle and far-reaching conjecture of Fermat would be the equivalent, not in pages but in sheer labor and genius, of writing a fair-sized book in the humanities. But Cauchy was a tireless man, with more than 700 such papers to his name when he died. In any case, the strain and the bad air at Cherbourg wore on his health, so he left the engineering works to become a professor at the École Polytechnique, at age twenty-two.

Politically Incorrect

Throughout his professorial career, Augustin-Louis Cauchy was willing to pay for his convictions. His unabashed Catholicism grated on the nerves of his secular colleagues. When the Bourbon dynasty fell in the revolution of 1830, Cauchy did not wait for his colleagues to act, but went into exile along with the king. The next ten years or so found him in Turin, in Prague, then back in Paris. The Italians created a chair in mathematical physics just for him; the French liberals would not be s generous. When Cauchy applied for a position at the Bureau of Longitudes, he was voted down because he would not renounce the legitimacy of the Bourbon kings. For one of his applications to an ordinary post he was rejected by the academicians, forty-two to three. Imagine Einstein applying for a job as an assistant professor at Princeton — or Rutgers — and gaining only three out of forty-five votes. For several years, Cauchy was not even earning any money, and by then he was a married man with a wife and two daughters to support.

Throughout his professorial career, Augustin-Louis Cauchy was willing to pay for his convictions.

When the revolution of 1848 brought Napoleon III to the throne, the new king exempted Cauchy from taking the loyalty oath, and so the great mathematician could live out the rest of his days in France. But that didn't mean that he kept silent about important matters. The French secularists wanted to banish the Church from public life, consigning it to the walls of your home or your parish church. Call it "freedom of worship," 19th-century version, with rigorous restraints upon the free exercise of faith. Cauchy would have none of it. He stood up for the Jesuits when they were attacked. He stood up for the independence of their schools. He stood up for the preservation of the Lord's Day as a day free of labor. He wrote to the pope on behalf of the Irish during the potato famine. He was personally active in works of charity as a member of the Society of Saint Vincent de Paul.

Cauchy was also a close friend of the most potent Jesuit preacher in France, Gustave Xavier de Ravignan. One longs to have been a witness to their conversations: the mathematician and the Jesuit priest, known for his ascetic life, his zeal for souls, and his precise logic in defense of the faith. Of Père de Ravignan they say that his writings are wanting in flights of imagination, but he was a powerful and magnetic presence. Thousands of men flocked to his retreats. I think there is a certain kind of man, pure, exact, relentlessly logical, who is moved less by emotional outbursts than by the brilliant light of reasoning from premises to conclusions. Cauchy was one such, and evidently there were many others.

How Firm a Foundation

Augustin-Louis Cauchy, France's finest mathematician, was an ardent Catholic. What influence did his faith have on his mathematical work? I'll end this essay with something that mathematicians might appreciate: a conjecture.

For God has created the world — not an abstract game, but the world — in measure, weight, and number. Augustin-Louis Cauchy saw it, and believed.

Cauchy wrote in the introduction to his most important work, the Corns d'analyse de l'École royale polytechnique (1821), that mathematicians before him had committed "sins," and that he was going to correct them. One writer suggests that this language of sin and confession comes out of the Council of Trent. But we do not need to invoke Trent; and there's a better way to understand what Cauchy wanted.

Cauchy had in mind two tendencies among his predecessors. One was to rely on demonstrations that fell short of airtight proof. The other was to consider algebra as a system of abstractions dealing with other abstractions, a self-enclosed language that referred ultimately only to itself. The first "sin" was like a flawed attempt to prove the existence of God; a mind like that of Thomas Aquinas — or Cauchy — would not be satisfied until the proof was rectified.

The second "sin" has implications that are profounder still. Cauchy always insisted that mathematics does in fact refer to really existing things, even if the things are but geometrical figures, like a triangle or a polyhedron. He wanted to give a firm basis to the discoveries made by his predecessors, both by cleaning up their proofs and by referring them back, again and again, to realities — though perhaps realities that can be perceived only by the mind.

For God has created the world — not an abstract game, but the world — in measure, weight, and number. Augustin-Louis Cauchy saw it, and believed.

Acknowledgement

Anthony Esolen. "How the Church Has Changed the World: "Measure, Weight, and Number." Magnificat (August, 2017).

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